1) Используем формулы понижения степени:
sin^2 x = (1 - cos 2x)/2, cos^2 x = (1 + cos 2x)/2
Тогда
int sin^4 x dx = int (sin^2 x)^2 dx = int ((1 - cos 2x)/2)^2 dx =
= int (1 - cos 2x)^2/4 dx = 1/4 * int (1 - 2 * cos 2x + cos^2 2x) dx =
= 1/4 * int (1 - 2 * cos 2x + (1 + cos 4x)/2) dx =
= 1/4 * int (1 - 2 * cos 2x + 1/2 + 1/2 * cos 4x) dx =
= 1/4 * int (3/2 - 2 * cos 2x + 1/2 * cos 4x) dx =
= 1/4 * (3/2 * x - 2 * 1/2 * sin 2x + 1/2 * 1/4 * sin 4x) + C =
= 1/4 * (3/2 * x - sin 2x + 1/8 * sin 4x) + C =
= 3/8 * x - 1/4 * sin 2x + 1/32 * sin 4x + C
2) y = sin x, y = 1, x = 0
S = int (0 pi/2) (1 - sin x) dx = (x + cos x)_{0}^{pi/2} =
= (pi/2 + cos pi/2) - (0 + cos 0) = pi/2 - 1.
Ответ: S = pi/2 - 1.